This subproject is one of many research subprojects utilizing the resources provided by a Center grant funded by NIH/NCRR. Primary support for the subproject and the subproject's principal investigator may have been provided by other sources, including other NIH sources. The Total Cost listed for the subproject likely represents the estimated amount of Center infrastructure utilized by the subproject, not direct funding provided by the NCRR grant to the subproject or subproject staff. In single-particle electron microscopy (EM), the reconstruction of an electron density map depends fundamentally upon the assignment of Euler angles that describe the orientation of corresponding 2D projection images. A correct assignment will enable proper back-projection of the images in 3D space to create an appropriate volume, while an incorrect assignment will produce various degrees of aberrations. Thus, the challenge is identifying the true Euler angles for all projections. The problem can be solved either by directly tilting the specimen inside the microscope or through the identification of angular relationships within the 2D images themselves (for a comprehensive review, see (Frank, 2006a)). When the 2D images represent at least several views of homogeneous macromolecular complexes on the EM grid and can be described by well-defined contrast-transfer function (CTF) values, they contain all the necessary information for Euler angle assignment without requiring additional macroscopic tilting. According to the central section theorem, the 2D Fourier transform of a projection image represents a slice (central section) through the 3D Fourier transform of the corresponding object. The 2D transforms are in turn represented as radial lines in Fourier space, at least one of which will necessarily be shared by any pair of central sections, while their exact number is uniquely dictated by the object's symmetry. Proper identification (or approximation) of common lines for all combinations of images will allow for the determination of angular relationships between 2D projections and the assignment of Euler angles to each, which would enable subsequent 3D reconstruction.